There is well known a vehicle having an automatic transmission configured to transmit a drive force from a drive power source toward drive wheels. Patent Documents 1-3 disclose examples of such type of vehicle. Generally, this type of vehicle is generally is arranged to implement a basic shift control of the automatic transmission. The basic shift control is implemented to determine or select one of gear positions (operating positions, speed positions) or gear ratios (speed ratios) of the automatic transmission, on the basis of a state of the vehicle as represented by a running speed of the vehicle and an operation amount of an accelerator pedal (a manually operated vehicle accelerating member, and according to a relation such as a shifting map which is stored in a memory and which represents a predetermined relationship between the vehicle state and the gear position (or gear ratio) of the automatic transmission that is to be selected or established. The automatic transmission is shifted to the selected gear position (or gear ratio) to establish the corresponding speed ratio. In addition to this basic shift control, the automatic transmission is controlled to be shifted down from the presently established gear position, upon detection of a so-called “kick-down operation” of the accelerator pedal by a kick-down operation detecting switch, for example. This shift-down operation of the automatic transmission is unnecessary if the acceleration of the vehicle required by the vehicle operator who has kicked down the accelerator pedal can be realized without the shift-down operation, for instance, if the vehicle is running down on a downhill roadway. Patent Document 1 discloses a control technique to compare the required vehicle acceleration value represented by the operation amount of the accelerator pedal with an estimated vehicle acceleration value to be achieved as a result of the kick-down operation of the accelerator pedal, and to inhibit the unnecessary shift-down operation of the automatic transmission if the estimated vehicle acceleration value reaches the required vehicle acceleration value.